International Mathematical Olympiad 1988

Consider two coplanar circles of radii $R$ and $r$ ($R > r$) with the same center. Let $P$ be a fixed point on the smaller circle and $B$ a variable point on the larger circle. The line $BP$ meets the larger circle again at $C$. The perpendicular $l$ to $BP$ at $P$ meets the smaller circle again at $A$. (If $l$ is tangent to the circle at $P$ then $A = P$.)

(i) Find the set of values of $BC^2 + CA^2 + AB^2$.

(ii) Find the locus of the midpoint of $BC$.

Let $n$ be a positive integer and let $A_1$, $A_2$, \ldots, $A_{2n+1}$ be subsets of a set $B$. Suppose that

(a) Each $A_i$ has exactly $2n$ elements,

(b) Each $A_i \cap A_j$ ($1 \leq i < j \leq 2n + 1$) contains exactly one element, and

(c) Every element of $B$ belongs to at least two of the $A_i$.

For which values of $n$ can one assign to every element of $B$ one of the numbers $0$ and $1$ in such a way that $A_i$ has $0$ assigned to exactly $n$ of its elements?

A function $f$ is defined on the positive integers by

$$\begin{aligned} f(1) &= 1, \quad f(3) = 3, \\ f(2n) &= f(n), \\ f(4n + 1) &= 2f(2n + 1) - f(n), \\ f(4n + 3) &= 3f(2n + 1) - 2f(n), \end{aligned}$$

for all positive integers $n$.

Determine the number of positive integers $n$, less than or equal to 1988, for which $f(n) = n$.

Show that set of real numbers $x$ which satisfy the inequality

$$\sum_{k=1}^{70} \frac{k}{x-k} \geq \frac{5}{4}$$

is a union of disjoint intervals, the sum of whose lengths is 1988.

$ABC$ is a triangle right-angled at $A$, and $D$ is the foot of the altitude from $A$. The straight line joining the incenters of the triangles $ABD$, $ACD$ intersects the sides $AB$, $AC$ at the points $K$, $L$ respectively. $S$ and $T$ denote the areas of the triangles $ABC$ and $AKL$ respectively. Show that $S \geq 2T$.

Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^2 + b^2$. Show that

$$\frac{a^2 + b^2}{ab + 1}$$

is the square of an integer.